Music: A Mathematical Offering

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Since the time of the Ancient Greeks, much has been written about the relationship between mathematics and music: from harmony and number theory, to musical patterns and group theory. Benson provides a wealth of information here to enable the teacher, the student or the interested amateur to understand, at varying levels of technicality, the real interplay between these two ancient disciplines. The story is long as well as broad, and involves physics, biology, psychoacoustics, the history of science and digital technology as well as, of course, mathematics and music. Starting with the structure of the human ear and its relationship with Fourier analysis, the story proceeds via the mathematics of musical instruments to the ideas of consonance and dissonance, and then to scales and temperaments. This is a must-have book if you want to know about the music of the spheres or digital music and many things in between.
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Editorial Reviews

From the Publisher
"Perhaps our children will one day remark on the group symmetries in their favorite music in the way that we now simply note a beautiful tune. They, no less than we, will have much to learn from this delightful book, which sets a new standard of excellence and inclusiveness. Anyone who knows some college-level mathematics and is curious about how it can illuminate music will be richly rewarded by reading Benson's outstanding book."
Peter Pesic, Tutor and Musician-in-Residence at St. John's College, Santa Fe

"... A precise selection of solutions..."
Luigi Carlo Berselli, Mathematical Reviews

"... an excellent introduction to the interdisciplinary subject of music and mathematics (which also involves physics, biology, psycho-acoustics, and the history of science and digital technology). The book can easily be used as the text for undergraduate courses."
The Mathematical Intelligencer

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Product Details

  • ISBN-13: 9780521853873
  • Publisher: Cambridge University Press
  • Publication date: 8/31/2006
  • Edition description: First Edition
  • Pages: 426
  • Product dimensions: 6.85 (w) x 9.72 (h) x 0.98 (d)

Meet the Author

David J. Benson is a 6th Century Professor in the Department of Mathematical Sciences at the University of Aberdeen.

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Table of Contents

Preface     xi
Acknowledgements     xiii
Introduction     1
Waves and harmonics     5
What is sound?     5
The human ear     7
Limitations of the ear     13
Why sine waves?     17
Harmonic motion     18
Vibrating strings     19
Sine waves and frequency spectrum     21
Trigonometric identities and beats     23
Superposition     26
Damped harmonic motion     28
Resonance     31
Fourier theory     36
Introduction     37
Fourier coefficients     38
Even and odd functions     44
Conditions for convergence     46
The Gibbs phenomenon     50
Complex coefficients     54
Proof of Fejer's theorem     55
Bessel functions     58
Properties of Bessel functions     61
Bessel's equation and power series     63
Fourier series for FM, feedback and planetary motion     68
Pulse streams     71
The Fourier transform     73
Proof of the inversion formula     77
Spectrum     80
The Poisson summation formula     81
The Dirac delta function     82
Convolution     86
Cepstrum     88
The Hilbert transform and instantaneous frequency     89
A mathematician's guide to the orchestra     91
Introduction     91
The wave equation for strings     92
Initial conditions     100
The bowed string     103
Wind instruments     107
The drum     112
Eigenvalues of the Laplace operator     117
The horn     120
Xylophones and tubular bells     122
Thembira     130
The gong     133
The bell     138
Acoustics     142
Consonance and dissonance     144
Harmonics     144
Simple integer ratios     145
History of consonance and dissonance     148
Critical bandwidth     151
Complex tones     152
Artificial spectra     153
Combination tones     155
Musical paradoxes     158
Scales and temperaments: the fivefold way     161
Introduction      162
Pythagorean scale     162
The cycle of fifths     164
Cents     165
Just intonation     167
Major and minor     168
The dominant seventh     170
Commas and schismas     171
Eitz's notation     172
Examples of just scales     174
Classical harmony     181
Meantone scale     185
Irregular temperaments     189
Equal temperament     198
Historical remarks     202
More scales and temperaments     210
Harry Partch's 43 tone and other just scales     210
Continued fractions     214
Fifty-three tone scale     223
Other equal tempered scales     227
Thirty-one tone scale     228
The scales of Wendy Carlos     231
The Bohlen-Pierce scale     233
Unison vectors and periodicity blocks     237
Septimal harmony     242
Digital music     245
Digital signals     245
Dithering     247
WAV and MP3 files     248
MIDI     251
Delta functions and sampling     251
Nyquist's theorem      254
The z-transform     256
Digital filters     257
The discrete Fourier transform     261
The fast Fourier transform     263
Synthesis     265
Introduction     265
Envelopes and LFOs     266
Additive synthesis /     268
Physical modelling     270
The Karplus-Strong algorithm     273
Filter analysis for the Karplus-Strong algorithm     275
Amplitude and frequency modulation     276
The Yamaha DX7 and FM synthesis     280
Feedback, or self-modulation     287
CSound     291
FM synthesis using CSound     298
Simple FM instruments     300
Further techniques in CSound     304
Other methods of synthesis     308
The phase vocoder     309
Chebyshev polynomials     309
Symmetry in music     312
Symmetries     312
The harp of the Nzakara     322
Sets and groups     324
Change ringing     329
Cayley's theorem     331
Clock arithmetic and octave equivalence     333
Generators      335
Tone rows     337
Cartesian products     339
Dihedral groups     340
Orbits and cosets     342
Normal subgroups and quotients     343
Burnside's lemma     345
Pitch class sets     348
Polya's enumeration theorem     353
The Mathieu group M12     358
Bessel functions     361
Equal tempered scales     365
Frequency and MIDI chart     367
Intervals     368
Just, equal and meantone scales compared     372
Music theory     374
Recordings     381
References     386
Bibliography     389
Index     393
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